John Guckenheimer

Research Overview

I engage in research on dynamical systems and their
applications. Even the simplest
nonlinear
dynamical systems can generate phenomena of bewildering
complexity. Formulas that describe
their trajectories seldom exist, so computer
simulations are invaluable in understanding their behavior.
Theoretical advances have been inspired by common patterns observed
while simulating many different systems. One of the main
goals of my research is to discover these patterns and characterize
their properties. The resulting theory then serves as a guide in
studying the dynamics of specific systems. It is also the foundation
for numerical algorithms that seek to analyze
system
behavior in ways that go beyond simulation.

My research is a blend of theoretical
investigation,
development
of computer methods and studies of nonlinear systems
that arise
in diverse fields of science and engineering. Two of the primary themes
have been bifurcation theory, which studies the
dependence of dynamical behavior upon system parameters, and the
effects of multiple time scales in shaping dynamical behavior.
Application areas in which I have worked include population biology,
fluid dynamics, neurosciences, animal
locomotion and control of nonlinear systems. My
work on algorithm development includes contributions to methods
for computing bifurcations, periodic orbits and invariant manifolds of
vector fields and for the analysis of fractal dimensions of attractors.